Abstract:Let R" be n-dimensional Euclidean space, and let c~: [0, L] -~ R" and fl: [0, L] -, ~" be closed rectifiable arcs in N" of the same total length L which are parametrized via their arc length. fl is said to be a chord-stretched version of • if for each 0 ~< s ~< t ~< L, Is(t)-e(s)l ~< Ifl(t) -fl(s)l, fl is said to be convex if fl is simple and if fl([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if ~ were simple then there existed a … Show more
“…We say that γ majorizes γ provided that there exists a nonexpanding map f : C → X with f • γ = γ. The curve γ has also been called an "unfolding" [15,25] or "chord-stretching" [40,46] of γ. We call f the majorization map.…”
We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature M n , n ≥ 2, is bounded below by the volume of the unit sphere in Euclidean space R n . This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in M , via Kleiner's variational approach, and thus settles the Cartan-Hadamard conjecture. The proof employs a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for smooth approximation of the signed distance function. Immediate applications include sharp extensions of the
“…We say that γ majorizes γ provided that there exists a nonexpanding map f : C → X with f • γ = γ. The curve γ has also been called an "unfolding" [15,25] or "chord-stretching" [40,46] of γ. We call f the majorization map.…”
We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature M n , n ≥ 2, is bounded below by the volume of the unit sphere in Euclidean space R n . This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in M , via Kleiner's variational approach, and thus settles the Cartan-Hadamard conjecture. The proof employs a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for smooth approximation of the signed distance function. Immediate applications include sharp extensions of the
“…See also [6,8] and the references therein. More results on edge-length preserving transformations and chord stretching can be found in [4,12,13,14].…”
Section: Introductionmentioning
confidence: 99%
“…Observe that for the operation to be well-defined we need that pi−1 and pi+1 are distinct. This operation belongs to the larger class of edge-length preserving transformations, when applied to polygons [4,12,13,14]. It seems to have been used for the first time by Millet.…”
A length preserving transformation of a polygon is any transformation of its vertices that preserves the lengths of the edges. In the video segment we will demonstrate three types of length preserving transformations: pocket flips, flipturns, and pops. We present sequences of such operations and study their power (or weakness) in the attempt of convexifying simple polygons.
“…Observe that for the operation to be well-defined we need that p i−1 and p i+1 are distinct. This operation belongs to the larger class of edge-length preserving transformations, when applied to polygons [5,19,20,21,22]. It seems to have been used for the first time by Millet [16].…”
Section: Introductionmentioning
confidence: 99%
“…There is an extensive bibliography pertaining to these subjects [1,2,3,4,5,6,8,9,10,12,13,14,15,17,18,23,24,25]. See also [5,19,20,21,22] for more results on edge-length preserving transformations and chord stretching.…”
Given a polygon P in the plane, a pop operation is the reflection of a vertex with respect to the line through its adjacent vertices. We define a family of alternating polygons, and show that any polygon from this family cannot be convexified by pop operations. This family contains simple, as well as non-simple (i.e., self-intersecting) polygons, as desired. We thereby answer in the negative an open problem posed by Demaine and O'Rourke [9, Open Problem 5.3].
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